A Calabi-Yau variety of dimension d is a complex manifold with trivial
canonical bundle and vanishing Hodge numbers h i,0 for 0
< i < d. For instance, a dimension 1 Calabi-Yau variety is an elliptic
curve, a dimension 2 Calabi-Yau variety is a K3 surface, and a dimension
3 is a Calabi-Yau threefold.

(A) One of the most significant developments in the last decade in
Theoretical Physics (High Energy) is, arguably, string theory
and mirror symmetry. String theory proposes a model for the physical
world which purports its fundamental constituents as 1-dimensional mathematical
objects "strings" rather than 0- dimensional objects "points".
Mirror symmetry is a conjecture in string theory that certain "mirror
pairs" of Calabi-Yau manifolds give rise to isomorphic physical
theories. Calabi-Yau manifolds appear in the theory because in passing
from the 10-dimensional space time to a physically realistic description
in four dimension, string theory requires that the additional 6-dimensional
space is to be a Calabi-Yau manifold.

Though the idea of mirror symmetry has originated in physics, in recent
years, the field of mirror symmetry has exploded onto the mathematical
scene. It has inspired many new developments in algebraic geometry,
toric geometry, Riemann surfaces theory, infinite dimensional Lie algebras,
among others. For instance, mirror symmetry has been used to tackle
the problem of counting the number of rational curves on Calabi-Yau
threefolds.

In the course of mirror symmetry, it has become more apparent that
Calabi-Yau varieties enjoy tremendously rich arithmetic properties.
For instance, arithmetic objects such as: modular forms, modular functions
of one and more variables, algebraic cycles, L-functions, and p-adic
L-functions, have popped up onto the scene. Also special classes
of Calabi-Yau manifolds, e.g., of Fermat type hypersurfaces, or their
deformations pertinent to mirror symmetry, offer promising testing grounds
for physical predictions as well as rigorous mathematical analysis and
computations.

(B) One of the most significant developments in the last decade in
Arithmetic Geometry and Number Theory is the proof of the Taniyama-Shimura-Weil
conjecture of the so-called modularity of elliptic curves defined
over the field of rational numbers by A. Wiles and his disciples.
Wiles' idea is to exploit 2-dimensional Galois representations arising
from elliptic curves and modular forms of weight 2 on some congruence
sup-groups of PSL(2,Z), and establish their equivalence. His method
ought to be applied to explore arithmetic of Calabi-Yau threefolds.
In particular, rigid Calabi-Yau threefolds defined over the field of
rational numbers are equipped with 2-dimensional Galois representations,
which are conjecturally equivalent to modular forms of one variable
of weight 4 on some congruence subgroup of PSL(2, Z). For not necessarily
rigid Calabi-Yau threefolds over the rationals, the Langlands Program
predicts that there should be some automorphic forms attached to them.
We plan to test the so-called modularity conjectures for Calabi-Yau
varieties defined over the field of rational numbers, or more generally,
over number fields, first trying to understand them for some special
classes of Calabi-Yau threefolds, e.g., those mentioned in (A).

(C) There are a number of intriguing developments in the theory of
algebraic cycles in the past 25 years, that not surprisingly, should
open the door to an infusion of new techniques in the study of Calabi-Yau
manifolds and mirror symmetry. The impact of classical Hodge theory
as well as the p-adic Hodge cycles, is clearly evident. On the algebraic
side, there is the relationship of algebraic K-theory and Chow groups
of algebraic cycles, leading to the Bloch-Quillen-Gersten resolution
description of Chow groups. There is also the more recent relationship
of Bloch's higher Chow groups and higher K-theory (a higher Riemann-Roch
theorem), and a conjectured "arithmetic index theorem". The influence
of the work of Bloch and Beilinson on the subject of algebraic cycles
is profound. For instance there are the fascinating Bloch-Beilinson
conjectures on the existence of a natural filtration on the Chow groups,
whose graded pieces can be described in terms of extension data, and
their conjectures about injectivity of certain regulators of cycle groups
of varieties over number fields. There is also the work of others on
how conjecturally this filtration can be explained in terms of kernels
of higher regulators and arithmetic Hodge structures. The Calabi-Yau
manifolds present an ideal testing ground for some of these conjectures.

2. Objectives
The recent progress mentioned above (A), (B) and (C), based on so many
interactions with so may areas of mathematics and physics, have contributed
to a considerable degree of inaccessibility to mathematicians and physicists
working in their respective fields, not to mention, graduate students.
Perhaps one of the greatest obstacles facing mathematicians and physicists
is that each camp has its own language. Mathematicians have had difficulty
isolating mathematical ideas in physics literatures, and vice versa
for physicists. In recent years, however, these barriers have started
melting away with enormous efforts by both camps. Several summer schools
and workshops are planned in the hope of narrowing these gaps, to name
a few, "The Geometry of Supergravity" at IAS/Park City Summer Session,
2001 and "The Duality Workshop: A Math/Physics Collaboration" at Institute
for Theoretical Physics at University of California Santa Barbara, 2001.
Our workshop will inevitably have some overlaps, however, we are hoping
that ours has a distinctively arithmetic favour, and complementary to
the other workshops.

Geometry around mirror symmetry and string theory has been pursued
by many mathematicians (complex geometers, toric geometers, and others),
and great progress has been witnessed in understanding geometric aspects
of the problem. In fact, recently a number of excellent books and survey
articles have been published explaining complex geometric aspects of
mirror symmetry on Calabi-Yau threefolds as well as on K3 surfaces.

Further, in the past two decades, a number of people who have studied
that part of algebraic geometry dealing with Hodge theory and algebraic
cycles, have found applications of their work in Quantum Cohomology,
Mirror Symmetry and Calabi-Yau manifolds. One anticipates that these
interactions between the various "schools" will blossom in the near
future.

Arithmetic aspects on Calabi-Yau varieties and mirror symmetry, however,
are yet to be explored vigorously. For instance, Wiles' method should
be explored to establish the modularity for rigid Calabi-Yau threefolds
defined over the field of rational numbers a la Fontaine and Mazur.
Also, investigation on the intermediate Jacobians of Calabi-Yau threefolds
ought to be pursued using, for instance, p-adic Hodge theory. Recent
articles of P. Candelas et.al. on the computation of the zeta-functions
of Calabi-Yau manifolds over finite fields reveal a surprising connection
of mirror symmetry to p-adic L-functions (which are the essential ingredients
in Iwasawa theory).
Further investigation on p-adic analysis in physics (pertinent to mirror
symmetry to begin with) ought to be carried out.

The construction of algebraic cycles on Calabi-Yau threefolds (generalizing
the method of Bloch), investigation of L-functions of Calabi-Yau threefolds
a la the conjectures of Beilinson and Bloch, among others, ought to
be pursued with more rigour and intensity.

Our goal is to bring together to the Fields Institute experts, recent
Ph.D.'s and graduate students, working in physics, geometry and arithmetic
around Calabi-Yau varieties and mirror symmetry, and to exchange ideas
and learn the subjects first-hand mingling with researchers with different
expertise. We expect these interactions to lead to progress in solving
open problems in mathematics and physics as well as to pave way to new
developments.

3. Expected participants
Mathematicians and physicists in Canadian institutions who are interested
in the workshop are all welcome. From outside Canada, the following
mathematicians and theoretical physicists have confirmed their participation
in the workshop as of June 28, 2001:

4. Proceedings.
A proceedings of the workshop is planed to be published from the Fields
Institute Communication Series. The publisher of the series is the American
Mathematica
Society. All the necessary informations about preparing manuscript
(in LaTex) are attached to the workshop booklet or may be found here.